Cloudinary's WordPress plugin indeed supports multi-sites.
- Use a different Cloudinary account for each site.
- Share the same Cloudinary account for all the sites and set a different folder in your Cloudinary storage path for each website.
For both options, you should (for each WordPress site) go to the Connect tab under the Cloudinary in the left menu of your WordPress admin interface, and enter the
CLOUDINARY_URL as appears in the dashboard of Cloudinary's Management Console. In case you chose option 2, you will also need to go to the Sync Media tab under the Cloudinary and add a folder to the Cloudinary storage path.